Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-252x\)
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(homogenize, simplify) |
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\(y^2z=x^3-252xz^2\)
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(dehomogenize, simplify) |
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\(y^2=x^3-252x\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-6, 36)$ | $0.60597056133774074777658573606$ | $\infty$ |
| $(18, 36)$ | $0.89250642214408953269148501916$ | $\infty$ |
| $(0, 0)$ | $0$ | $2$ |
Integral points
\((-14,\pm 28)\), \((-12,\pm 36)\), \((-6,\pm 36)\), \((-3,\pm 27)\), \( \left(0, 0\right) \), \((16,\pm 8)\), \((18,\pm 36)\), \((21,\pm 63)\), \((42,\pm 252)\), \((64,\pm 496)\), \((84,\pm 756)\), \((162,\pm 2052)\), \((450,\pm 9540)\), \((1050,\pm 34020)\)
Invariants
| Conductor: | $N$ | = | \( 14112 \) | = | $2^{5} \cdot 3^{2} \cdot 7^{2}$ |
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| Discriminant: | $\Delta$ | = | $1024192512$ | = | $2^{12} \cdot 3^{6} \cdot 7^{3} $ |
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| j-invariant: | $j$ | = | \( 1728 \) | = | $2^{6} \cdot 3^{3}$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z[\sqrt{-1}]\) (potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $N(\mathrm{U}(1))$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $0.41839793624631896313891784248$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.3105329259115095182522750833$ |
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| $abc$ quality: | $Q$ | ≈ | $$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $2.951599931990628$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.52538491582749474772839718972$ |
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| Real period: | $\Omega$ | ≈ | $1.3162005887018343630441260353$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2^{2}\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $5.5320954840576982502384767431 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.532095484 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.316201 \cdot 0.525385 \cdot 32}{2^2} \\ & \approx 5.532095484\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 8192 |
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{3}^{*}$ | additive | -1 | 5 | 12 | 0 |
| $3$ | $4$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $7$ | $2$ | $III$ | additive | -1 | 2 | 3 | 0 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.
| $\ell$ | Reduction type | Serre weight | Serre conductor |
|---|---|---|---|
| $2$ | additive | $2$ | \( 63 = 3^{2} \cdot 7 \) |
| $3$ | additive | $6$ | \( 1568 = 2^{5} \cdot 7^{2} \) |
| $7$ | additive | $20$ | \( 288 = 2^{5} \cdot 3^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 14112.c
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 1568.f1, its twist by $12$.
The minimal quartic twist of this elliptic curve is 32.a3, its quartic twist by $12348$.
Growth of torsion in number fields
The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{7}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.0.12348.1 | \(\Z/4\Z\) | not in database |
| $8$ | 8.0.2439569664.6 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.624529833984.14 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.2.16862305517568.1 | \(\Z/6\Z\) | not in database |
| $8$ | 8.0.78066229248000.3 | \(\Z/10\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/10\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/10\Z\) | not in database |
We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | add | ord | add | ss | ord | ord | ss | ss | ord | ss | ord | ord | ss | ss |
| $\lambda$-invariant(s) | - | - | 2 | - | 2,4 | 2 | 2 | 2,2 | 2,2 | 2 | 2,2 | 2 | 2 | 2,2 | 2,2 |
| $\mu$-invariant(s) | - | - | 0 | - | 0,0 | 0 | 0 | 0,0 | 0,0 | 0 | 0,0 | 0 | 0 | 0,0 | 0,0 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.